Marshak_Elektron_tabl

Nmgdp•y ϕ (x ijbcfZ} \b]ey^ ϕ [ = [ − I [ • •l_jZp•cgZ nhjfmeZ

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[Q = [Q− − I [Q− , n= 1,2,3...

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aZ[_ai_qm} a[•`gbc ijhp_k

AZ gmevh\_ gZ[eb`_ggy x0 fh`gZ ijbcgylb Z[h agZq_ggy gZ h^ghfm •a

[ = D + E Z[h

agZq_ggy \ [m^v-ydbc lhqp• \•^j•adm \bagZq_g_ a i_\gbo f•jdm\Zgv

Ydsh qbkeh r \bagZqZ}lvky •a mfh\b _ϕ'(x)| ” r < 1 lh djbl_j•}f aZ\_j- r_ggy •l_jZp•cgh]h mlhqg_ggy dhj_gy •a aZ^Zghx lhqg•klx ε } mfh\Z

_ [ − [Q _≤ ε − U .

U

>ey ijZdlbqgbo h[qbke_gv \ ydhkl• qbkeZ U h[bjZxlv gZc[•evr_ agZ- q_ggy i_jrh€ iho•^gh€ nmgdp•€ ϕ'(x gZ ijhf•`dm •aheyp•€ dhj_gy lh[lh r =

= max |ϕ'(x _ ijb a ≤ x ≤ b.

AZagZqbfh sh •kgmxlv lZdh` •gr• f_lh^b gZcr\b^rh]h kimkdm ?cld_- gZ-Kl_nn_gk_gZ <_]kl_cgZ Jb[Zdh\Z • l ^ ^ey mlhqg_ggy dhj_g•\ yd• fZxlv \bkhdm r\b^d•klv a[•`ghkl•

lhqg•klx ε = 10 - 4.

Jha\¶yahd Ijb\_^_fh j•\gyggy ^h aZ]Zevgh]h \b]ey^m f (x) = 0.

[ − FRV [ − = , I [ = [ − FRV [ − .

1gl_j\Ze •aheyp•€ dhj_gy6 [0,5; 1,5] >ey mlhqg_ggy dhj_gy ijb\_^_fh j•\gyggy f (x) = 0

^h \b]ey^m x = ϕ (x ^_ ϕ (x) = x – f (x).

ϕ [ = [ − I [ = [ − [ − FRV [ − = [ − [ + FRV [ + .

G_h[o•^gh i_j_\•jblb djbl_j•c a[•`ghkl• f_lh^Z •l_jZp•c Ydsh nmgdp•y ϕ (x \bagZ- q_gZ • fZ} g_i_j_j\gm iho•^gm gZ ^_ydhfm •gl_j\Ze• ijbqhfm ϕ′ [ < lh py ihke•^h\g•klv

a[•]Z}lvky ^h dhj_gy j•\gyggy x = ϕ (x gZ pvhfm •gl_j\Ze•

AgZc^_fh ϕ ′ [ = − − VLQ [ .

[

6 1gl_j\Ze •aheyp•€ fh`_ [mlb \bagZq_gbc [m^v-ydbf •a \bs_ aZagZq_gbo kihkh[•\

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ϕ′ [ < gZ •gl_j\Ze• •aheyp•€ dhj_gy [0,5; 1,5].

LZdbf qbghf •l_jZp•cgbc ijhp_k [m^_ a[•`gbf

>Ze• ^ey mlhqg_ggy dhj_gy j•\gyggy f_lh^hf ijhklh€ •l_jZp•€ a lhqg•klx ε = 10 - 4 aZ

^hihfh]hx _e_dljhggbo lZ[ebpv g_h[o•^gh aZih\gblb dhf•jdb lZ[ebqgh]h ijhp_khjZ gZklmigbf qbghf ihke•^h\g•klv hi_jZp•c h^gZdh\Z • ^ey MS Excel • ^ey OO Calc).

< dhf•jdm K \\_klb lhqg•klv agZoh^`_ggy dhj_gy

>Ze• ihlj•[gh aZih\gblb rZidm lZ[ebp• jbk

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Ihl•f m dhf•jdm A7 ke•^ \\_klb agZq_ggy – e•\m ]jZgbpx •gl_j\Zem •aheyp•€ dh-

Ijhp_k mlhqg_ggy dhj_gy ke•^ aZ\_jrblb lh^• dheb mfh\Z _ [ − [ _≤ ε − U [m^_

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U

\bdhgZgZ >ey pvh]h h[qbkebfh U = PD[ _ ϕ ′ [ _ ijb a ≤ x ≤ b. Nmgdp•y gZ ijhf•`dm >a, b]

fhghlhggZ lhfm fh`gZ h[qbkeblb agZq_ggy fh^mey i_jrh€ iho•^gh€ nmgdp•y gZ d•gpyo •gl_j\Zem •aheyp•€ _ ϕ′ D _ lZ _ ϕ′ - _ lZ h[jZlb gZc[•evr_ agZq_ggy

<\_^_fh m dhf•jdm G nhjfmem $%6- DHJ?GV(E3))-6,1 ( m dhf•jdm G4

^ey OO Calc: =IF(ABS(A7-A6)<$C$3*(1-( (Dhj•gv o 5281'$

Ihl•f ihlj•[gh \b^•eblb ^•ZiZahg : D lZ ijhly]gmlb aZ fZjd_j aZiheg_gby ^h lbo i•j ihdb g_ [m^_ agZc^_gh dhj•gv jbk

Jbk 18

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GZ[eb`_gbc ^h^Zlgbc dhj•gv j•\gyggy [ − FRV [ = ydbc agZc^_gh f_lh^hf ijhklh€ •l_jZp•€ [ ≈ .

IjbdeZ^ 10 Mlhqgblb h^bg a dhj_g•\ j•\gyggy x 2 − cos x = 0 f_lh^hf ijhklh€ •l_jZp•€ a lhqg•klx ε = 10 - 4.

Jha\¶yahd. I [ = [ − FRV [ 1gl_j\Ze •aheyp•€ dhj_gy [0,8 0,9].

>ey mlhqg_ggy dhj_gy ijb\_^_fh j•\gyggy f (x) = 0 ^h \b]ey^m x = ϕ(x ^_ ϕ(x)= x – f(x). ϕ(x)= x – f(x) = x – ( x2 – cos x) = x – x2 + cos x.

I_j_\•jbfh djbl_j•c a[•`ghkl• f_lh^Z •l_jZp•c lh[lh qb \bdhgm}lvky mfh\Z |ϕ'(x)|<1

gZ •gl_j\Ze• •aheyp•€ dhj_gy > @

AgZc^_fh _ϕ'(x)| = 1-2x – sin x. |ϕ'(0,8)|=|1-2*0,8-sin(0,8)| § ! |ϕ'(0,9)|=|1-2*0,9-sin(0,9)| § !

|ϕ'(x)|>1gZ •gl_j\Ze• •aheyp•€ dhj_gy

LZdbf qbghf •l_jZp•cgbc ijhp_k [m^_ jha[•`gbf

>ey lh]h sh[ aZ[_ai_qblb a[•`g•klv ijhp_km ihfgh`bfh h[b^\• qZklbgb j•\gyggy

AZa\bqZc gZ ijZdlbp• h[bjZxlv M • max|f'(x _ gZ \•^j•adm >a, b]. f' (x)=2x+sin x.

| f' (0,8)|=2*0,8+sin (0,8)§ | f' (0,9)|=2*0,9+sin (0,9)§

Hl`_ fh`gZ h[jZlb 0 = j•\gyggy fZlbf_ \b]ey^ [ − FRV [ =

I_j_\•jbfh mfh\m a[•`ghkl• f_lh^Z •l_jZp•c ^ey hljbfZgh]h j•\gyggy

I [ = [ − FRV [ ,

ϕ [ = [ − I [ = [ − [ − FRV [ .

ϕ′ [ = − I ′ [ = − [ + VLQ [ .

ϕ′ = − + VLQ ≈ < ,

ϕ′ = − + VLQ ≈ < .

LZdbf qbghf •l_jZp•cgbc ijhp_k [m^_ a[•`gbf

>ey mlhqg_ggy dhj_gy j•\gyggy x2 – cos x gZ •gl_j\Ze• •aheyp•€ > @ f_lh^hf ijhklh€ •l_jZp•€ a lhqg•klx ε = 10 - 4 aZ ^hihfh]hx _e_dljhggbo lZ[ebpv g_h[o•^gh aZih\gblb dhf•jdb lZ[ebqgh]h ijhp_khjZ yd hibkZgh m IjbdeZ^• ihke•^h\g•klv hi_jZp•c h^gZdh\Z • ^ey MS Excel • ^ey OO Calc).

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\bdhjbklh\mxlvky \ lbo \biZ^dZo dheb j_amevlZl h[qbke_gv fZ} [mlb gZc[eb`qbf [•evrbf Z[h gZc[eb`qbf f_grbf p•ebf qbkehf m dhf•jdm F8 =1-F7/F6.

>ey \b\_^_ggy agZq_ggy dhj_gy \ dhf•jdm ' ke•^ \\_klb nhjfmem –

^ey MS Excel: ?KEB $%6$ -A6)<$C$3*(1-$F$8)/$F Dhj•gv o HDJM=E $

^ey OO Calc: =IF(ABS(A7-A6)<$C$3*(1-$F$8)/$F Dhj•gv o 5281'$

Ihl•f ihlj•[gh \b^•eblb ^•ZiZahg : D lZ ijhly]gmlb aZ fZjd_j aZiheg_gby ^h lbo i•j ihdb g_ [m^_ agZc^_gh dhj•gv jbk

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<[m^h\Zgbc •gkljmf_gl Ih^[hj iZjZf_ljZ. Jha\¶yaZggy [Z]Zlvho ijZd-

lbqgbo hkh[eb\h •g`_g_jgbo aZ^Zq ihlj_[m} agZoh^`_ggy qbkeh\bo agZq_gv iZjZf_lj•\ nmgdp•c sh aZ[_ai_qmxlv hljbfm\Zggy hq•dm\Zgbo agZq_gv nmgd-

p•€ Yd ijZ\beh p_ fZxlv [mlb _dklj_fZevg• \_ebqbgb Z[h dhgdj_lg• agZq_ggy 1kgmxlv \[m^h\Zg• •gkljmf_glb _e_dljhggbo lZ[ebpv yd• ih\bgg• kijh-

sm\Zlb dhjbklm\Zqm ihrmd pbo iZjZf_lj•\ gZijbdeZ^ Ih^[hj iZjZf_ljZ. Ih^[hj iZjZf_ljZ – p_ •gkljmf_gl ydbc ^ha\hey} p•e_kijyfh\Zgh i_-

j_[jZlb [Z]Zlh agZq_gv h^bghqgh]h iZjZf_ljm a h^ghqZkgbf dhgljhe_f d•gp_- \h]h agZq_ggy nmgdp•€

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ih\bgg_ f•klblb Z^j_km Z[h •f¶y dhf•jdb ydZ f•klblv nhjfmem ^h ydh€ ke•^ i•^•[jZlb iZjZf_lj

AgZq_gb_ (P_e_\h_ agZq_gb_) – p_ ihe_ ih\bggh f•klblb agZq_ggy yd_

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ydh€ ih\bggh ^hky]gmlb [Z`Zgh]h dhgdj_lgh]h Z[h _dklj_fZevgh]h agZq_ggy

Baf_gyy agZq_gb_ yq_cdb (Baf_gy_fZy yq.) – ih\bggh f•klblb Z^j_km

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JHA< YA:GGY KBKL?F E1G1CGBO :E=?;J:2QGBO J1<GYGV

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